A342034 - OEIS

A342034

a(n) is the number of numbers k with n digits where k has digits in nondecreasing order and satisfies k < (product of digits of k) * (sum of digits of k).

8, 41, 140, 367, 789, 1432, 2276, 3280, 4326, 5350, 6254, 7009, 7588, 7970, 8175, 8210, 8120, 7923, 7633, 7272, 6877, 6445, 6013, 5555, 5122, 4693, 4298, 3901, 3534, 3189, 2872, 2562, 2285, 2029, 1789, 1576, 1376, 1194, 1037, 893, 759, 654, 548, 454, 384, 315, 254, 210, 168, 127, 97, 79, 56, 39, 31, 21, 12, 8, 4

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OFFSET

1,1

COMMENTS

As A066310 is finite there exists m such that a(n) = 0 for all n > m.

a(n) = 0 for n >= 85 since 9^n*9n <= 10^(n-1) for n >= 85. This may occur as early as n = 60, as 9^n*9n <= 10^n-1 for n >= 60. But a(59) > 0 since 10^59-1 < 9^59*9*59. - Michael S. Branicky, Mar 05 2021

LINKS

Table of n, a(n) for n=1..59.

David A. Corneth, PARI program

EXAMPLE

a(1) = 8 as there are 8 one-digit numbers k as described in name. Those are {2, 3, 4, 5, 6, 7, 8, 9}.

PROG

(Python)

from math import prod

from itertools import combinations_with_replacement as cwr

def c(digs): return int("".join(map(str, digs))) < prod(digs) * sum(digs)

def a(n): return sum(1 for u in cwr(range(1, 10), n) if c(u))

print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Mar 05 2021